Preheat the oven to 375 degrees F. Grease a 10-by-1 1/2-inch round baking dish with the butter and sprinkle with 1 tablespoon of granulated sugar.
In a small bowl, combine the prunes with 2 tablespoons of Armagnac and microwave on high for 1 minute. Set aside.
In the bowl of an electric mixer fitted with the paddle attachment, beat the eggs and 1/3 cup granulated sugar on medium-high speed for 3 minutes, until light and thick. In a small bowl, combine the flour, baking powder, and salt. Combine the cream, vanilla, lemon zest, and 1 tablespoon Armagnac in a glass measuring cup. On low speed, slowly add the dry ingredients to the batter and then add the liquid ingredients, mixing well. Set aside for 10 minutes
Distribute the prunes in the prepared pan and carefully pour on the batter. Sprinkle evenly with 1 teaspoon granulated sugar. Bake for 35 to 40 minutes, until the top is golden brown and a toothpick comes out clean. Cool slightly, sprinkle with confectioners' sugar, and serve warm.